Qiang Zhen, Charles Knessl
We consider a processor shared $M/M/1$ queue that can accommodate at most a finite number $K$ of customers. We give an exact expression for the sojourn time distribution in the finite capacity model, in terms of a Laplace transform. We then give the tail behavior, for the limit $K\to\infty$, by locating the dominant singularity of the Laplace transform.
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We consider the $M/M/1$ queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison \cite{MO} and Flatto \cite{FL}.
Explicit formulas for the moments of the sojourn time in the M/G/1 processor sharing queue with permanent jobs
介绍了近年来在处理器共享规则（EPS）下分析M/G/1队列方面取得的一些成果。导出了j阶矩（j= 1，2，…）M/G/1--EPS排队逗留时间中K（K= 0，1，2，…）无限大的永久性工作。并讨论了如何简化矩的计算。
Sojourn times in the M/G/1 FB queue with light-tailed service times
研究了前台-后台服务规则下平稳M/G/1排队系统中顾客逗留时间的渐近衰减率。FB原则为那些迄今为止获得最少服务的客户提供服务。我们证明了对于轻尾服务时间，逗留时间的衰减率等于忙期的衰减率。它表明，FB最小化的衰变率在类的工作保存学科。
We study the asymptotics of the stationary sojourn time Z of a "typical customer" in a tandem of single-server queues. It is shown that, in a certain "intermediate" region of light-tailed service time distributions, Z may take a large value mostly due to a large value of a single service time of one of customers. Arguments used in the paper allow us to obtain also an elementary proof of the logarithmic asymptotics for the tail distribution of the stationary sojourn time in the whole class of light-tailed distributions.
We consider a single-server GI/GI/1 queueing system with feedback. We assume the service times distribution to be (intermediate) regularly varying. We find the tail asymptotics for a customer's sojourn time in two regimes: the customer arrives in an empty system, and the customer arrives in the system in the stationary regime. In particular, in the case of Poisson input we use the branching processes structure and provide more precise formulae. As auxiliary results, we find the tail asymptotics for the busy period distribution in a single-server queue with an intermediate varying service times…
We consider a finite population processor-sharing (PS) queue, with Markovian arrivals and an exponential server. Such a queue can model an interactive computer system consisting of a bank of terminals in series with a central processing unit (CPU). For systems with a large population $N$ and a commensurately rapid service rate, or infrequent arrivals, we obtain various asymptotic results. We analyze the conditional sojourn time distribution of a tagged customer, conditioned on the number $n$ of others in the system at the tagged customer's arrival instant, and also the unconditional distributi…
We study the sojourn time in a queueing system with a single exponential server, serving a Poisson stream of customers in order of arrival. Service is provided at low or high rate, which can be adapted at exponential inspection times. When the number of customers in the system is above a given threshold, the service rate is upgraded to the high rate, and otherwise, it is downgraded to the low rate. The state dependent changes in the service rate make the analysis of the sojourn time a challenging problem, since the sojourn time now also depends on future arrivals. We determine the Laplace tran…
Asymptotic Expansions for the Sojourn Time Distribution in the $M/G/1$-PS Queue
我们考虑具有处理器共享服务器的$M/G/1$队列。研究了以顾客服务需求为条件的条件逗留时间分布和无条件分布在各种渐近极限下的情形。这些包括大时间和/或大服务请求，以及高负荷服务，其中到达速率仅略小于服务速率。我们的研究结果证明了无条件分布可能的尾部行为，这是以前已知的情况下$G=M$和$G=D$（其中它是纯指数）。我们假设服务密度至少以指数速度衰减。我们使用各种方法的渐近展开的积分，如拉普拉斯和鞍点方法。
In this paper, we analyze the sojourn of an entire batch in a processor sharing $M^{[X]}/M/1$ processor queue, where geometrically distributed batches arrive according to a Poisson process and jobs require exponential service times. By conditioning on the number of jobs in the systems and the number of jobs in a tagged batch, we establish recurrence relations between conditional sojourn times, which subsequently allow us to derive a partial differential equation for an associated bivariate generating function. This equation involves an unknown generating function, whose coefficients can be com…
In this paper, we exploit results obtained in an earlier study for the Laplace transform of the sojourn time $Ω$ of an entire batch in the $M^{[X]}/M/1$ Processor Sharing (PS) queue in order to derive the asymptotic behavior of the complementary probability distribution function of this random variable, namely the behavior of $P(Ω>x)$ when $x$ tends to infinity. We precisely show that up to a multiplying factor, the behavior of $P(Ω>x)$ for large $x$ is of the same order of magnitude as $P(ω>x)$, where $ω$ is the sojourn time of an arbitrary job is the system. From a practical point of view, t…
Higher-Order Approximations of Sojourn Times in M/G/1 Queues via Stein's Method
通过Stein方法，研究M/G/1队列中驻留时间的高阶近似，利用矩匹配条件提升指数近似精度。
We consider the steady-state distribution of the sojourn time of a job entering an M/GI/1 queue with the foreground-background scheduling policy in heavy traffic. The growth rate of its mean, as well as the limiting distribution, are derived under broad conditions. Assumptions commonly used in extreme value theory play a key role in both the analysis and the results.
For the $M^{[X]}/M/1$ processor Sharing queue with batch arrivals, the sojourn time $Ω$ of a batch is investigated. We first show that the distribution of $Ω$ can be generally obtained from an infinite linear differential system. When further assuming that the batch size has a geometric distribution with given parameter $q \in [0,1[$, this differential system is further analyzed by means of an associated bivariate generating function $(x,u,v) \mapsto E(x,u,v)$. Specifically, denoting by $s \mapsto E^*(s,u,v)$ the one-sided Laplace transform of $E(\cdot,u,v)$ and defining $$ Φ(s,u,v) = P(s,u) \…
We consider sojourn or response times in processor-shared queues that have a finite population of potential users. Computing the response time of a tagged customer involves solving a finite system of linear ODEs. Writing the system in matrix form, we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large. This corresponds to finite population models where the total population is $N\gg 1$. Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Hermite equation. The dominant eigenvalue leads to the t…
Sojourns of fractional Brownian motion queues: transient asymptotics
研究分数布朗运动驱动的排队过程在高阈值之上的逗留时间渐近性，推导布朗运动情形的精确渐近和一般情形的锐渐近近似，并分析两种比率机制。
We consider sojourn (or response) times in processor-shared queues that have a finite customer capacity. Computing the response time of a tagged customer involves solving a finite system of linear ODEs. Writing the system in matrix form, we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large. This corresponds to finite capacity models where the system can only hold a large number $K$ of customers. Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Airy equation. The dominant eigenvalue leads…
On Sojourn Times in the $M/M/1$-PS Model, Conditioned on the Number of Other Users
我们考虑处理器共享的$M/M/1$ -PS队列。我们研究了到达顾客的条件逗留时间分布，条件是其他顾客的数量。利用离散格林函数，得到了条件逗留时间分布的一个新公式。这证明了它与Pollaczeck和Vaulot在1946年的一些经典结果是等价的。然后，各种渐近极限进行了研究，包括大的时间和/或大量的客户存在，和高负荷服务，到达率仅略小于的服务率。
This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruit…
Communication networks need to support voice and data calls simultaneously. This results in a queueing system with heterogeneous agents. One class of agents demand immediate service, would leave the system if not provided. The second class of customers have longer job requirements and can wait for their turn. We discuss the achievable region of such a two class system, which is the region of all possible pairs of performance metrics. Blocking probability is the relevant performance for eager/impatient class while the expected sojourn time is appropriate for the second tolerant class. We obtain…
We consider a cyclic polling system with general service times, general switch-over times, and simultaneous batch arrivals. This means that at an arrival epoch, a batch of customers may arrive simultaneously at the different queues of the system. For the locally-gated, globally-gated, and exhaustive service disciplines, we study the batch sojourn-time, which is defined as the time from an arrival epoch until service completion of the last customer in the batch. We obtain for the different service disciplines exact expressions for the Laplace-Stieltjes transform of the steady-state batch sojour…
We consider the processor sharing $M/M/1$-PS queue which also models balking. A customer that arrives and sees $n$ others in the system "balks" (i.e., decides not to enter) with probability $1-b_n$. If $b_n$ is inversely proportional to $n+1$, we obtain explicit expressions for a tagged customer's sojourn time distribution. We consider both the conditional distribution, conditioned on the number of other customers present when the tagged customer arrives, as well as the unconditional distribution. We then evaluate the results in various asymptotic limits. These include large time (tail behavio…
We study a single server queue under a processor-sharing type of scheduling policy, where the weights for determining the sharing are given by functions of each job's remaining service(processing) amount, and obtain a fluid limit for the scaled measure-valued system descriptors.
Processor Sharing Queues with Impatience
研究了一类服务时间服从一般分布的处理机共享更新排队系统.不耐烦的工作可能会在完成服务之前放弃队列，或反悔。相应的随机过程表示为R^2+上的测度值马氏过程.一个缩放过程，产生一个非平凡的，但易处理的稳态行为的流体模型，提出。这个流体模型模型捕捉了底层随机模型的许多基本特征，它被用来分析处理器共享队列中的不耐烦的影响。
Stability of a processor sharing queue with varying throughput
在本文中，我们提出了一个稳定的标准处理器共享队列，其中的吞吐量可能取决于系统中的客户数（在这种情况下，如用户之间的干扰）。这样一个系统是由一个点测量值的随机递归跟踪客户的剩余处理时间。
The Foregound-Background Processor-Sharing Queue: a survey
前景-背景（FB）纪律，这给客户提供服务，已收到最少的服务量，最小化队列长度为一定类别的重尾服务时间。在本文中，我们给出了一个概述的结果，在文献中的单服务台队列与FB纪律。
Diffusion approximation for a processor sharing queue in heavy traffic
考虑一个具有更新到达和i.i.d.的单服务器队列。服务器在处理器共享服务规程下操作的服务时间。为了描述这个系统的演变，我们使用一个测度值的过程，在任何给定的时间跟踪系统中的所有作业的剩余服务时间。从这个度量值过程中，可以恢复传统的性能过程，包括队列长度和工作负载。我们表明，在温和的假设下，包括标准的高负荷服务的假设，（适当重新调整）的测量值的过程对应的一个序列的处理器共享队列收敛分布的测量值的扩散过程。极限过程的特征在于一个适当的提升映射下的图像，一维反射布朗运动。作为一个直接的结果，一个处理器共享队列的队列长度过程的扩散近似。
A large-deviations analysis of the GI/GI/1 SRPT queue
研究了具有最短剩余处理时间和轻尾服务时间的GI/GI/1排队系统。我们的兴趣集中在sojourn-time分布的尾部行为。我们得到了它的大偏差衰减率的一般表达式。这个衰减率的值主要取决于服务时间分布的终点是否有质量。辅助优先排队在我们的分析中起着重要的作用，我们得到了一些新的结果。我们应用我们的SRPT结果比较SRPT与FIFO从大偏差的角度来看。
In this short communication we study a fluid queue with a finite buffer. The performance measure we are interested in is the occupation time over a finite time period, i.e., the fraction of time the workload process is below some fixed target level. We construct an alternating sequence of sojourn times $D_1,U_1,...$ where the pairs $(D_i,U_i)_{i\in\mathbb{N}}$ are i.i.d. random vectors. We use this sequence to determine the distribution function of the occupation time in terms of its double transform.
Invariant states and rates of Convergence for a critical fluid model of a processor sharing queue
本文对重负载处理机共享队列的流体模型进行了渐近分析。特别地，我们考虑了临界流体模型的解在时间接近\infty时的行为。本文的主要定理为流体模型解收敛到不变状态提供了充分条件，并且在稍微更严格的假设下，提供了一个收敛速率。Gromoll的相关工作中使用这些结果建立了一个处理器共享队列的高负荷服务扩散近似。
The diffusion approximation of the Multiclass Processor Sharing queue
本文研究带反馈的多类处理器共享队列，在重负载条件下，为追踪作业剩余服务时间的测度值过程建立了扩散近似。
Fluid and Diffusion Approximation of a Multiclass Proportional Processor Sharing Queue with Reneging and General Distributions
本文解决了多类比例处理器共享队列（含弃权（不执行）及一般分布）的扩散近似问题，通过推广流体极限并证明扩散缩放下的紧性与唯一性，刻画了由高斯片驱动的随机微分方程系统。
Tempered Erlang Queue with Multiple Arrivals
针对多重到达 Erlang 排队系统，本文利用缓增稳定从属过程的首次击中时间进行时间变换，建立并求解分数阶差分-微分方程，给出状态概率、等价阶段表示、平均队长及多种关键时间分布。
We consider a sequence of single-server queueing models operating under a service policy that incorporates batches into processor sharing: arriving jobs build up behind a gate while waiting to begin service, while jobs in front of the gate are served according to processor sharing. When they have been completed, the waiting jobs move in front of the gate and the cycle repeats. We model this system with a pair of measure valued processes describing the jobs in front of and behind the gate. Under mild asymptotically critical conditions and a law-of-large-numbers scaling, we prove that the pair o…
Heavy traffic limit for a processor sharing queue with soft deadlines
本文研究了一个GI/GI/1处理机共享队列，其中的作业具有软截止期。在每个时间点，剩余的服务时间和最后期限的收集建模使用右半平面上的随机计数措施。在扩散标度和高负荷服务条件下得到了该测度值过程的极限，并刻画为极限队列长度过程的确定性函数.作为特殊情况，得到了提前时间曲线和排队时间曲线的扩散近似。还得到了逗留时间的快拍原理。
Consider a single server system serving a multiclass population. Some popular scheduling policies for such system are the discriminatory processor sharing (DPS), discriminatory random order service (DROS), generalized processor sharing (GPS) and weighted fair queueing (WFQ). In this paper, we propose two classes of policies, namely MPS (multiclass processor sharing) and MROS (multiclass random order service), that generalize the four policies mentioned above. For the special case when the multiclass population arrive according to Poisson processes and have independent and exponential service r…
In this paper we study a class of Quasi-Skipfree (QSF) processes where the transition rate submatrices in the skipfree direction have a column times row structure. Under homogeneity and irreducibility assumptions we show that the stationary distributions of these processes have a product form as a function of the level. For an application, we will discuss the ${\it Cox(k)}/M^Y/1$-queue, that can be modelled as a QSF process on a two-dimensional state space. In addition we study the properties of the stationary distribution and derive monotonicity of the mean number of the customers in the queu…
We consider a processor sharing queue where the number of jobs served at any time is limited to $K$, with the excess jobs waiting in a buffer. We use random counting measures on the positive axis to model this system. The limit of this measure-valued process is obtained under diffusion scaling and heavy traffic conditions. As a consequence, the limit of the system size process is proved to be a piece-wise reflected Brownian motion.
We consider a tandem queue with coupled processors, which is subject to global breakdowns. When the network is in the operating mode and both queues are non empty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the non-empty station. Moreover, arrival rates depend on the state of the network. The system is described by a Markov modulated random walk in the quarter plane representing the number of jobs in the two stations and the state of the network. By applying the generatin…
We compute the stationary performance metrics of a single server $M^X/G/1$ queue under a class of generalized processor-sharing scheduling policies that are proposed by Grishechkin. This class of processor-sharing policies allow service capacities to be allocated to jobs based on the amount of service they attained. In [11], Grishechkin derives an integral equation that is satisfied by the Laplace transform of the stationary performance metrics under these policies. Our main focus in this paper is to derive the solution to this integral equation. This is achieved through a series of transforms…
Response time in a pair of processor sharing queues with Join-the-Shortest-Queue scheduling
针对采用JSQ调度的并行PS队列响应时间分布求解难题，本文利用生成函数法推导条件响应时间的PDE，通过数值求解与复积分去条件化得到分布解，结果与50万周期再生模拟吻合良好。
We study the convergence of the $M/G/1$ processor-sharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to Lévy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward excursions obtained from those of some reflected Brownian motion with drift, after taking the image of their local time process by the Lamperti transformation. We also show, via excursion theoretic arguments, that this entails the convergence of the entire processes to …
A two-class Processor-Sharing queue with one impatient class is studied. Local exponential decay rates for its stationary distribution (N, M) are established in the heavy traffic regime where the arrival rate of impatient customers grows proportionally to a large factor A. This regime is characterized by two time-scales, so that no general Large Deviations result is applicable. In the framework of singular perturbation methods, we instead assume that an asymptotic expansion of the solution of associated Kolmogorov equations exists for large A and derive it in the form P(N = Ax, M = Ay) ~ g(x,y…
Fork-join network is a class of queueing networks with applications in manufactory, healthcare and computation systems. In this paper, we develop a simulation algorithm that (1) generates i.i.d. samples of the job sojourn time, jointly with the number of waiting tasks, exactly following the steady-state distribution, and (2) unbiased estimators of the derivatives of the job sojourn time with respect to the service rates of the servers in the network. The algorithm is designed based on the Coupling from the Past (CFTP) and Infinitesimal Perturbation Analysis (IPA) techniques. Two numerical exam…
The paper considers a queueing system with limited processor sharing. No more than n jobs may be served simultaneously. This system may be used for modeling bandwidth sharing in wireless communication systems and processes of service in computer networks. If there are n jobs in the considered queueing system and a new job arrives, then the arriving job is lost or the service of a job is interrupted and this job is lost. We study two rules to choose the job to be lost. In accordance with one of these rules, the job with the shortest remaining length is lost. Relations are obtained between the s…
We consider a stochastic network with mobile users in a heavy-traffic regime. We derive the scaling limit of the multi-dimensional queue length process and prove a form of spatial state space collapse. The proof exploits a recent result by Lambert and Simatos which provides a general principle to establish scaling limits of regenerative processes based on the convergence of their excursions. We also prove weak convergence of the sequences of stationary joint queue length distributions and stationary sojourn times.
We consider an overloaded multiclass nonidling first-in-first-out single-server queue with abandonment. The interarrival times, service times, and deadline times are sequences of independent and identically, but generally distributed random variables. In prior work, Jennings and Reed studied the workload process associated with this queue. Under mild conditions, they establish both a functional law of large numbers and a functional central limit theorem for this process. We build on that work here. For this, we consider a more detailed description of the system state given by $K$ finite, nonne…
We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/$\infty$-type queue. A job that is not completed during a visit will have a new service time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, …
We analyze randomized dynamic load balancing schemes for multi-server processor sharing systems when the number of servers in the system is large and the servers have heterogeneous service rates. In particular, we focus on the classical power-of-two load balancing scheme and a variant of it in which a newly arrived job is assigned to the server having the least instantaneous Lagrange shadow cost among two randomly chosen servers. The instantaneous Lagrange shadow cost at a server is given by the ratio of the number of unfinished jobs at the server to the capacity of the server. Two different a…
Strategic Customer Behavior in an M/M/1 Feedback Queue with General Payoffs
本文研究 M/M/1 反馈队列中客户基于阈值策略的策略性加入行为，在服务奖励随时间折现的设定下，证明悖论效应仍可能出现，并开发数值方法恢复逗留时间分布以推导均衡策略。
We consider an infinite-buffer single-server queue where inter-arrival times are phase-type ($PH$), the service is provided according to Markovian service process $(MSP)$, and the server may take single, exponentially distributed vacations when the queue is empty. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function (VGF) of system-length distribution at a pre-arrival epoch. Also, we obtain the steady-state system-length distribution at an arbitrary epoch along with some important performance measures such as the mean number of cus…
On the Age of Information of Processor Sharing Systems
本文研究处理器共享（PS）队列中信息年龄（AoI），推导M/M/1/2队列平均AoI闭式解，并扩展到M/M/1多源场景，发现PS在某些情况下优于M/M/1/1*。
We analyze the latency or sojourn time L(m,n) for the last customer in a batch of n customers to exit from the m-th queue in a tandem of m queues in the setting where the queues are in equilibrium before the batch of customers arrives at the first queue. We first characterize the distribution of L(m,n) exactly for every m and n, under the assumption that the queues have unlimited buffers and that each server has customer independent, exponentially distributed service times with an arbitrary, known rate. We then evaluate the first two leading order terms of the distributions in the large m and …
The single server queue with multiple customer types and semi-Markovian service times, sometimes referred to as the $M/SM/1$ queue, has been well-studied since its introduction by Neuts in 1966. In this paper, we apply an extension of this model, with batch arrivals and exceptional first service, to road traffic situations involving multiple streams of conflicting traffic. In particular, we use it in the context of gap acceptance models where low-priority traffic needs to cross (or, depending on the application, merge with) another traffic flow of higher priority. Traditionally, gap acceptance…
Two-Phase Phase-Type Queues: Closed-Form Distributions, Sensitivity Analysis and the Boundary of Algebraic Tractability
该研究推导了完整$M/PH_2/1$族队列的闭式分布，通过BuTools明确其数值精度格局，证明$PH_2$是代数可处理性边界，为两阶段服务模型提供了现成参考。
For a GI/GI/1 queue, we show that the average sojourn time under the (blind) Randomized Multilevel Feedback algorithm is no worse than that under the Shortest Remaining Processing Time algorithm times a logarithmic function of the system load. Moreover, it is verified that this bound is tight in heavy traffic, up to a constant multiplicative factor. We obtain this result by combining techniques from two disparate areas: competitive analysis and applied probability.
We consider a pair of coupled queues driven by independent spectrally-positive Levy processes. With respect to the bi-variate workload process this framework includes both the coupled processor model and the two-server fluid network with independent Levy inputs. We identify the joint transform of the stationary workload distribution in terms of Wiener-Hopf factors corresponding to two auxiliary Levy processes with explicit Laplace exponents. We reinterpret and extend the ideas of Cohen and Boxma (1983) to provide a general and uniform result with a neat transform expression.
By analyzing energy-efficient management of data centers, this paper proposes and develops a class of interesting {\it Group-Server Queues}, and establishes two representative group-server queues through loss networks and impatient customers, respectively. Furthermore, such two group-server queues are given model descriptions and necessary interpretation. Also, simple mathematical discussion is provided, and simulations are made to study the expected queue lengths, the expected sojourn times and the expected virtual service times. In addition, this paper also shows that this class of group-ser…
We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines distributions of random sums. With the help of the obtained results we analyse the distribution function of the time during which the Geiger-Muller counter will not lose any particles, the distribution function of the busy period of a redundant system with renewal, and the distribution function of the sojourn times of a single-server queueing system.
Queues with service resetting
针对服务时间由服务端减速与任务固有大小共同决定的S&X队列，构建服务重置的完整理论，分析两种主流重置策略下的系统性能，证明服务重置可有效提升该类排队系统性能。
Approximations and Optimal Control for State-dependent Limited Processor Sharing Queues
本文研究了服务速率依赖于占用服务器作业数的处理器共享服务器的近似与控制，提出基于重交通扩散分析的近优控制策略，通过构造状态依赖的处理器共享服务器序列获得漂移函数，并通过数值算法解决动态并发限制控制问题。
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